Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

TL;DR

This paper investigates how total positivity properties of Hankel matrices are preserved under sums, Hadamard products, and powers, providing sharp results and counterexamples that resolve key open problems.

Contribution

It establishes preservation conditions for total positivity in Hankel matrices under various operations and offers counterexamples demonstrating the sharpness of these results.

Findings

Total positivity is preserved under certain operations for Hankel matrices.

Counterexamples show the limits of these preservation properties.

Results resolve the Hadamard critical-exponent problem for specific matrix classes.

Abstract

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices.